Ground state selection under pressure in the quantum pyrochlore magnet Yb2Ti2O7

Abstract

A quantum spin liquid is a state of matter characterized by quantum entanglement and the absence of any broken symmetry. In condensed matter, the frustrated rare-earth pyrochlore magnets Ho₂ Ti₂ O₇ and Dy₂ Ti₂ O₇ , so-called spin ices, exhibit a classical spin liquid state with fractionalized thermal excitations (magnetic monopoles). Evidence for a quantum spin ice, in which the magnetic monopoles become long range entangled and an emergent quantum electrodynamics arises, seems within reach. The magnetic properties of the quantum spin ice candidate Yb₂ Ti₂ O₇ have eluded a global understanding and even the presence or absence of static magnetic order at low temperatures is controversial. Here we show that sensitivity to pressure is the missing key to the low temperature behaviour of Yb₂ Ti₂ O₇ . By combining neutron diffraction and muon spin relaxation on a stoichiometric sample under pressure, we evidence a magnetic transition from a disordered, non-magnetic, ground state to a splayed ferromagnetic ground state.

ARTICLE

Received 19 Jun 2016 | Accepted 31 Jan 2017 | Published 15 Mar 2017

E. Kermarrec1,2,3, J. Gaudet2, K. Fritsch4, R. Khasanov5, Z. Guguchia5, C. Ritter6, K.A. Ross7, H.A. Dabkowska8

& B.D. Gaulin2,8,9

A quantum spin liquid is a state of matter characterized by quantum entanglement and the absence of any broken symmetry. In condensed matter, the frustrated rare-earth pyrochlore magnets Ho2Ti2O7 and Dy2Ti2O7, so-called spin ices, exhibit a classical spin liquid state with fractionalized thermal excitations (magnetic monopoles). Evidence for a quantum spin ice, in which the magnetic monopoles become long range entangled and an emergent quantum electrodynamics arises, seems within reach. The magnetic properties of the quantum spin ice candidate Yb2Ti2O7 have eluded a global understanding and even the presence or absence of static magnetic order at low temperatures is controversial. Here we show that sensitivity to pressure is the missing key to the low temperature behaviour of Yb2Ti2O7. By combining neutron diffraction and muon spin relaxation on a stoichiometric sample under pressure, we evidence a magnetic transition from a disordered, non-magnetic, ground state to a splayed ferromagnetic ground state.

DOI: 10.1038/ncomms14810 OPEN

Ground state selection under pressurein the quantum pyrochlore magnet Yb2Ti2O7

1 Laboratoire de Physique des Solides, CNRS, Univ. Paris-Sud, Universit Paris-Saclay, Orsay Cedex 91405, France. 2 Department of Physics and Astronomy, McMaster University, Hamilton, Ontario, Canada L8S 4M1. 3 Laboratoire National des Champs Magntiques Intenses, CNRS, Grenoble BP 166-38042, France. 4 Helmholtz-Zentrum Berlin fr Materialien und Energie, Hahn-Meitner-Platz 1, Berlin 14109, Germany. 5 Laboratory for Muon Spin Spectroscopy, Paul Scherrer Institut, Villigen PSI CH-5232, Switzerland. 6 Institut Laue Langevin, BP 156, Grenoble 38042, France. 7 Department of Physics, Colorado State University, Fort Collins, Colorado 80523-1875, USA. 8 Brockhouse Institute for Materials Research, McMaster University, Hamilton, Ontario, Canada L8S 4M1. 9 Canadian Institute for Advanced Research, 180 Dundas St. W., Toronto, Ontario, Canada M5G 1Z8. Correspondence and requests for materials should be addressed to E.K. (email: mailto:[email protected]

Web End [email protected] ) or to B.D.G. (email: mailto:[email protected]

Web End [email protected] ).

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The pyrochlore lattice, comprised of corner-sharing tetrahedra, is the archetype of magnetic frustration in three dimensions1 (Fig. 1). Since its early study by

Anderson in 1956 (ref. 2), frustrated spin Hamiltonians on the pyrochlore lattice have provided a seemingly inexhaustible source for the study of fundamental physics3,4. In particular, spin liquid ground states have been predicted for such a lattice decorated with Heisenberg5,6 or XXZ7 spins. More recently, pyrochlore magnets have been put forward as realistic vehicles for the realization of a quantum spin ice state, using the generic S 12

nearest-neighbour anisotropic exchange Hamiltonian810. Yb2Ti2O7 is a promising quantum spin ice candidate as it possesses both an (effective) S 12 spin, thanks to the well isolated

crystal eld Kramers doublet ground state appropriate to Yb3 11, and strong quantum uctuations brought by anisotropic exchange interactions and an XY g-tensor12. Several studies have focused on the nature of the ground state in Yb2Ti2O7, yet a consensus has been elusive to date1318. Early neutron scattering experiments ruled out the presence of conventional static order down to 90 mK in a polycrystalline sample15, whereas other single crystal studies concluded the ground state was ferromagnetic14,16. The results of local probes are even more puzzling. Muon spin relaxation (mSR) measurements evidenced the presence of true static moments on the muon timescale, through the observation of both a drop of asymmetry and a decoupling of the muon spins in longitudinal applied elds18, along with a drastic slowing down of the uctuation rate below Tc for certain samples13. In contrast, mSR studies by DOrtenzio et al.17 found a non-magnetic, uctuating ground state, in both stoichiometric polycrystalline and single crystal samples, despite the presence of pronounced specic heat anomalies at Tc 265 mK and Tc 185 mK,

respectively. It is clear that local defects, either oxygen vacancies19 or excess magnetic ions20 (referred to as stufng), vary signicantly between polycrystalline powders and single crystals, and are likely responsible for such sample dependencies.

Here, by applying hydrostatic pressure to well-characterized Yb2 xTi2 xO7 d samples, with x 0 and x 0.046 (ref. 20), we

observe a magnetic transition in the stoichiometric, x 0 sample

from a disordered ground state into a splayed ferromagnetic ground state. This result sheds light on the origin of the sample dependence in the ground state selection for Yb2Ti2O7 and is consistent with the recent theoretical proposal that Yb2Ti2O7 lies close to a phase boundary in the generic quantum spin ice Hamiltonian phase diagram21.

ResultsMuon spin relaxation. mSR measurements under hydrostatic pressures as high as 25 kbar, and at temperatures as low as0.245 K, were performed on Yb2 xTi2 xO7 d samples, with

x 0 and x 0.046, at the GPD beamline of PSI. The muons

are implanted inside the bulk of the material, and act as local magnetic probes. The signal coming from the muons that stop inside the pressure cell was measured separately and subtracted (see Supplementary Figs 1 and 2) from the overall signal.

Figure 2a shows the temperature dependence of the mSR signal for the stoichiometric, x 0 sample in zero eld, Rzf(t), as a

function of time t and under an applied pressure P 19.7 kbar.

Well above Tc 0.265 K, at TZ0.97 K, the majority of the Yb3

magnetic moments are paramagnetic and in a fast uctuating regime, and display single-exponential relaxation. For Tr0.5 K, we observe the development of a small magnetic fraction f of the Yb3 moments, which grows nonlinearly as the temperature decreases. The absence of oscillations at short time is indicative of a highly disordered magnetic state. The zero-eld relaxation is well described by a Gaussian distribution of static internal elds with standard deviation D (see Supplementary Note 1), and the following phenomenological function:

Rzftf

2

3 e D

1 f e lt 1

In a purely static scenario, the second term (1/3-tail) should be constant. Here, a uctuating component is nonetheless observed and we modelled this using a relaxation rate l. The third term accounts for the paramagnetic component, and assumes the same relaxation rate l, for simplicity. The unconventional shape of the zero-eld longitudinal relaxation was discussed in detail in refs 13,22. In contrast, the evolution of the relaxation in temperature of the x 0 sample under zero applied pressure,

shown in Fig. 2b, shows little or no magnetic fraction (fC6%) at any temperature above our base T 0.245 K, in agreement with

DOrtenzio et al.17 previously reported mSR studies. Using equation (1) we extract the magnetic fraction for each pressure and temperature, and collect the results in Fig. 2c. The development of the magnetic fraction with temperature is clearly pressure dependent, and turns on strongly at low temperatures, below Tc 0.265 K, for our minimum pressure of

1.2 kbar. For each pressure, one can dene a critical temperature Tc, such that for TrTc, 50% of the magnetic moments are frozen.

The corresponding P T phase diagram is shown in Fig. 3.

Clearly, the phase transition extrapolated from nite pressure measurements to zero pressure agrees well with the sharp Cp anomaly at Tc 0.265 K, appropriate to the x 0 sample.

However the zero-pressure state for the x 0 sample at

0.245 K, below Tc, is disordered, indicating that the ground state of the stoichiometric, x 0 sample, is a spin liquid.

We now turn to the x 0.046 sample. The zero-eld relaxation

at T 0.245 K under zero and an applied pressure P 24.1 kbar

are shown in Fig. 2d. Strikingly, no frozen magnetic fraction is observed upon the application of a pressure as high as P 24.1

kbar. Instead, we observe an increase of the relaxation for this x 0.046 sample, demonstrating its sensitivity to pressure. The

temperature dependence of the relaxation is reported in Fig. 2e,f. One can speculate that a transition to a fully ordered state, as it is observed for the x 0 sample, would require higher pressures or

lower temperatures, consistent with the lower Tc 0.185 K of the

x 0.046 sample.

mSR studies on other samples have reported a drastic slowing down of spin uctuations13, or static order18, under zero applied pressure for temperatures below 0.25 K. In the light of our results, even relatively low (applied or chemical) pressure can destroy the

2t2=2

13 e lt

a b

x

y z

Figure 1 | Pyrochlore structure of Yb2 xTi2 xO7d. Excess Yb3 ion can

occupy a Ti4 site and create a local defect (referred to as stufng).(a) Representation of the ideal pyrochlore structure of Yb2Ti2O7, with Yb in blue, Ti in yellow and O in red. Yb and Ti form corner-sharing tetrahedra lattices. (b) Schematic representation of the structurally distorted environment of a defect.

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a b c

x = 0

1.0

0.8

0.6

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0.4

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4 5 0 1 2 3

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T (K)

0.6 0.7 0.8 0.9 1

4 5

d e f

1.0

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x = 0.046

x = 0.046

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T=0.245 KP =0 kbarP =24.1 kbar

0.8

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[afii9838](s1)

0.3

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0.1

0.0

4 0.2 0.4 0.6 0.8 1.0 1.2 1.4

5

0 1 2 3

t (s)

4 5 0 1 2 3 t (s)

Figure 2 | Temperature evolution of the lSR relaxation in Yb2xTi2 xO7 d under pressure. (ac) Refer to the x 0 and (df) refer to the x 0.046

sample. (a) The drastic increase of relaxation observed upon decreasing temperature in the x 0 sample indicates a spin freezing under an applied

pressure P 19.7 kbar, which is absent under zero pressure (b) and for the x 0.046 sample (d). (c) The temperature evolution of the magnetic fraction is

reported for various pressures. The black horizontal dashed line represents a volume magnetic fraction of 50%, used as a criterion to dene Tc. (e,f) For the x 0.046 sample, only a moderate increase of the spin dynamics is observed under applied pressure. The error bars of the mSR relaxation data are of

statistical origin and correspond to the square root of the total number of detected positrons resulting from muon decays. Error bars of the relaxation rate l represent standard deviation of the t parameters. Error bars of the magnetic fraction represent standard deviation of the t parameters, with a minimal value of 0.05 corresponding to the typical error on the total asymmetry for mSR under pressure.

40

10

P(kbar)

disordered spin liquid state and induce magnetic order. A low level of defects in the different samples is a natural explanation to the contradictory mSR results. Such disorder, at the B2% level, is difcult to characterize, but it is largely absent in polycrystalline samples, synthesized at lower temperatures by solid state methods.

Neutron diffraction. Armed with the knowledge of the P T

phase diagram in Fig. 3, we sought to determine the nature of the pressure-induced magnetic order in Yb2 xTi2 xO7 d

samples, with x 0, by performing neutron diffraction on the

stoichiometric powder sample at the D20 high-ux diffractometer of the ILL. The detection of small magnetic moments under pressure using neutron diffraction is challenging due to the signicant background signal of the pressure cell itself. Figure 4a shows the neutron diffraction data for the maximum hydrostatic pressure of the cell, P 11(2) kbar, and temperatures from 400 to

100 mK, from which a background measured at 800 mK was subtracted. We clearly observe the development of magnetic Bragg intensities at the (111), (311), (222) and (004) positions upon cooling below 400 mK. This is rm evidence for the existence of long-range magnetic order in Yb2 xTi2 xO7 d

samples, with x 0, under an applied pressure P 11(2) kbar.

The renement of the neutron diffraction data gives us the temperature dependence of the ordered moment, shown in Fig. 4b. The contrast with previous experiments under zero pressure is striking. First, the saturated moment m 0.33(5) mB is

1

0.0 0.1 0.2 0.3 0.4 0.5 T (K)

QSL

SFM

PM

Figure 3 | Pressuretemperature phase diagram of Yb2xTi2 xO7d.

The vertical axis displays the pressure P in logarithmic scale and the horizontal axis the temperature T. Empty black circles dene the transition line between the collective paramagnetic (PM, orange) and the splayed ferromagnetic (SFM, blue) regions relative to the x 0 sample. The

transition temperature is dened such that for TrTc 50% of the magnetic moments are frozen (see Fig. 2c). Error bars allow Tc to be dened between 40 and 60% of the magnetic fraction. The green region highlights the presence of a disordered, non-magnetic phase (QSL) found at P 0. Black

thick line is a guide to the eye. Dashed purple line is the hypothetical transition line for x 0.046.

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a b

0.35 1.0

0.8

0.6

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0.0

Magnetic fraction

5,400

(111) (200) (220) (311) (222) (400)

100 mK

200 mK

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Ordered moment ([afii9839] B)

0.15

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0.00

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T (K)

[afii9835]

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P = 12.2 kbar (SR) P = 11 kbar (neutron)

5,200

Intensity (a.u.)

5,000

4,800

4,600

c d

1 1.5 2 2.5

Q (1)

Figure 4 | Neutron diffraction measurements of Yb2Ti2O7 under applied pressure. (a) Diffraction data sets from 400 to 100 mK after the 800 mK data set has been subtracted. Error bars are not shown for clarity. (b) Ordered moment versus temperature determined by neutron diffraction for P 11(2) kbar

(blue, left axis) and magnetic fraction determined by mSR for P 12.2 kbar (orange, right axis). Error bars of the ordered moment represent s.d. of the

renement. Schematic spin structure of the ice-like splayed ferromagnet with y 14 for P 0 (c) and y 5 for P 11 kbar (d), where y is the splay angle

between the [001] direction and the magnetic moment, tilted towards the local [111] directions of the tetrahedron.

much smaller than that mB1 mB reported previously for different Yb2Ti2O7 samples16, although similar to the ordered moment in the G5 ordered state of Yb2Ge2O7 (ref. 23). Second, the ordered moment vanishes cleanly above TcB0.4 K, with no anomalous magnetic Bragg intensity well above Tc (refs 24,25). The previously reported order parameter at P 0 of our x 0

polycrystalline sample is anomalous24; it shows no change across Tc and only falls off at much higher temperatures.

Consistency with our P 0 mSR results on the same sample

requires that this Bragg-like scattering is dynamic on slow time scales. That notwithstanding, the magnetic structure previously rened on the basis of a very high temperature (B8 K)

background subtraction gave a splayed ice-like ferromagnetic structure24, with the moments on a tetrahedron lying mainly in the [100] direction with a positive splay angle y 14(5), such

that the moments tilt towards the local [111] direction (Fig. 4c). The components perpendicular to the local [111] axis obey the 2-in/2-out ice rule on a single tetrahedron. A different type of splayed ferromagnet, with the perpendicular components satisfying the all-in/all-out structure, has also been reported recently26, in addition to a nearly collinear ferromagnet (yB0)16, for other samples. The magnetic structure associated with the true Bragg scattering we rene here in the stoichiometric x 0 sample under P 11(2) kbar is also a splayed ice-like

structure, but with a much reduced splay angle y 5(4), such

that it is close to a collinear [100] ferromagnet (Fig. 4d).

DiscussionThese results bring a fresh perspective on the long standing debate about the presence or absence of static magnetic order in the quantum pyrochlore magnet Yb2 xTi2 xO7 d. The acute

sensitivity to local (through the Yb3 stufng) or applied pressure is surprising. However, a corollary of our new P T

phase diagram is that non-stoichiometric samples with non-zero chemical pressure can easily display an ambient applied pressure phase transition to a splayed ferromagnetic state at Tc. Yet, this interpretation remains challenged by the fact that our x 0.046

sample does not show evidence for magnetic order at ambient pressure, and by previous reported observations of a magnetic transition in polycrystalline, likely xB0, sample even under zero pressure13,18. This may indicate that the non-magnetic low-temperature region of the phase diagram is extremely

narrow, existing only for a certain range of x, whose absolute values are still to be determined. This would actually be reminiscent of the recent ndings on the Tb2 xTi2 xO7 d

pyrochlore magnet, which has been shown to display an ordered phase that is extremely sensitive to disorder, appearing only for 0oxo0.01 (refs 2729).

Furthermore, the present work illustrates the relevance of applying hydrostatic pressure to tune the magnetic properties of frustrated pyrochlore compounds, a path that was followed by pioneering work on the other spin liquid candidate Tb2Ti2O7 (ref. 30). In case of Yb2Ti2O7, we found that the pressure tunes the delicate balance between the anisotropic exchanges of the quantum spin ice Hamiltonian, and selects a splayed ferromagnetic ground state away from the degenerate antiferromagnetic ground states manifold. This scenario conrms recent theoretical proposals that Yb2Ti2O7 lies close to phase boundaries derived from the generic Seff 12 quantum spin

ice Hamiltonian21, and provides the missing key to understand its exotic magnetic properties. Particularly appealing is the prediction that accidental degeneracies in the vicinity of these phase boundaries can lead to the emergence of a quantum spin liquid31. This would offer a natural explanation for a non-magnetic, disordered state under zero pressure in stoichiometric Yb2Ti2O7 and recent observations of a continuum of gapless quantum excitations24,32 at low temperatures.

Methods

Sample preparation. The Yb2xTi2 xO7d samples with x 0 and x 0.046

were prepared at the Brockhouse Institute for Materials Research, McMaster University. The x 0 powder sample was obtained through conventional

solid-state reaction between pressed powders of Yb2O3 and TiO2 sintered at 1,200 C in air. The x 0.046 powder sample was obtained by crushing a single

crystal grown by the oating zone method in 4 atm of O2 with a growth rate of 5 mm h 1. More details on the details of the synthesis and the characterization can be found in ref. 25.

Muon spin relaxation. mSR measurements were carried out at the GPD instrument of the Paul Scherrer Institut, Switzerland. About 1 g of each powder sample was mixed with B3 mm3 of a pressure medium (Daphne 7373 oil) and placed inside the sample channel of a double-wall pressure cell. Two different cells were used, labelled as (1) and (2) (see Supplementary Note 1), and are described in more details in ref. 33. The muon momentum was adjusted in order to obtain an optimal fraction of the muons stopping in the sample, with optimal values found at P 106 and P 107 MeV c 1. The relaxation of both cells were measured without

any sample down to 0.245 K. The applied pressure was determined by measuring

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the superconducting transition temperature of a small piece of pure indium inserted in the sample channel33.

Neutron diffraction. The neutron diffraction experiment was conductedat the D20 beamline, a high intensity two axis diffractometer, at the Institut Laue-Langevin, using a neutron wavelength l 2.4 . A mass of 1.5 g of Yb2Ti2O7

powder sample and a small amount of NaCl powder, which serves as apressure calibration, were both mounted in a high pressure clamp cell and inserted in a 3He4He dilution fridge. Fluorinert was used as a pressure transmitter.

A minimum of 12 h of data was collected at each temperature. The diffraction pattern obtained for T 800 mK is shown in Supplementary Fig. 3. Structural

renements for both NaCl and Yb2Ti2O7 and magnetic renements for Yb2Ti2O7 have been performed using the Fullprof program suite34.

Data availability. The data sets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

Work at McMaster University was supported by NSERC of Canada. This work is based on experiments performed at SmS, Paul Scherrer Institute, Villigen, Switzerland and at the Institut Laue-Langevin, Grenoble, France. This project has received funding from the European Unions Seventh Framework Programme for research, technological development and demonstration under the NMI3-II Grant number 283883. E.K. acknowledges useful discussions with P. Mendels, F. Bert, S. Petit, L. D. C. Jaubert and C. Decorse.

Authors contributions

E.K., J.G. and B.D.G. wrote the manuscript. E.K., J.G., K.F. and B.D.G. performed the neutron diffraction experiment. E.K. and B.D.G. performed the mSR experiment.

K.A.R. and H.A.D. synthesized and characterized the samples. C.R. designed and performed the neutron scattering experiment. Z.G. and R.K. designed and performed the mSR experiment. All the co-authors discussed the results and improved the manuscript.

Additional information

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How to cite this article: Kermarrec, E. et al. Ground state selection under pressure in the quantum pyrochlore magnet Yb2Ti2O7. Nat. Commun. 8, 14810 doi: 10.1038/ncomms14810 (2017).

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r The Author(s) 2017

NATURE COMMUNICATIONS | 8:14810 | DOI: 10.1038/ncomms14810 | http://www.nature.com/naturecommunications

Web End =www.nature.com/naturecommunications 5